With the groundwork laid in the
last two chapters, we can now simulate the motion of electrons in the presence
of electric and magnetic fields. The right combination of E and B fields
will trap the electrons near the target surface, creating high electron densities
and consequently high sputter rates.

where F is the force
on the electron and q is the electron charge. We know from
NewtonÕs second law that force equals mass times acceleration. Using
this, along with the fact that the acceleration is the second derivative of
position, we can write equation (4- 1) as

(4- 2)

where
is the second derivative of position with time. This is a vector equation,
and for us it will be more convenient to write it as three equations, one for
each direction. If we donÕt remember the definition of a cross product,
Mathematica can remind us:

The electron equations of motion
are then

(4- 3)

(4- 4)

(4- 5)

where the first derivative of
position is used to represent the velocity components. By solving these
three coupled differential equations, we can determine the position of a single
electron over time. The Mathematica function NDSolve is designed just for
this type of problem. LetÕs try it out with a simplified B field, rather
than a full magnet array. Consider two long parallel magnets, as shown in
Figure 4- 1.

We will set each magnet to be 1 cm x 1cm x 40 cm
in size with a magnetization of Br = 1.4 T. Using our methodology from
Chapter 2, we can write this as

Code 4- 1. Definition of 2 magnet array.

With the B field functions from Chapter
2 (Error! Reference source not found.), we can plot the field. The
upper surface of the target is imagined to be at z=0, indicated by the
horizontal line in the figure. The voltage applied there will create our
E field.

From Equation Error! Reference
source not found., the field varies linearly with distance from the target.
In Mathematica we can write:

Code 4- 3. Definition of linear E field.

The E field is zero in the x and y
directions. In the Mathematica code for the z direction, we specify the
field using the Piecewise function. This allows us to distinguish the
sheath and nonsheath regions. Alternatively, an If statement can be used,
but this can cause problems when using NDSolve.

We will assume the target is set to -300 V and the
sheath is 1mm thick. Near the target, the electric field reaches -600,000
V/m, and then linearly drops to zero at the edge of the sheath.

Code 4- 4. Plotting the electric field.

Figure 4- 3. Linear electric field in
sheath.

We can now solve our differential equations by
calling the NDSolve function:

Code 4- 5. Solving the electron equations of motion.

The first portion of the NDSolve expression lists
our three equations of motion, for the x, y, and z directions. The next
section has the initial conditions for the electron. In our case these
are the starting position at the target surface (0, -0.007, 0) and the starting
velocity (0, 0, 0). Next we list the variables we are solving for (x, y,
z) and the range of times we are solving over. The last section has
various solver settings. These are used to guide the solver to find a
solution. In our case, boosting the accuracy and precision goals leads to
a more accurate solution.

It takes a few minutes to solve this
expression. Once it is done, Mathematica returns an interpolating
function for each variable. We can plot these using a parametric plot to
see the trajectory the electron takes over the target.

Code 4- 6. Electron trajectory plotted
using ParametricPlot

Figure 4- 4. Electron trajectory.

The electron follows a zigzag path,
drifting in the x direction. The side view of this trajectory can be seen
in Figure 4- 5, superimposed over the B field.

The electron leaves the target
surface almost vertically at y = -0.007 m and then begins circling around a B
field line until it reaches the target again. The E field then repels it
and it retraces its path to where it started. In this way, the electron
is trapped near the target surface. In addition to this back-and-forth
motion in the y-z plane, there is the ExB drift in x direction that takes the
electron down the magnetron.

To ensure our solution is accurate, we should
confirm that energy is conserved. At any given point the total energy of
the electron consists of its kinetic energy and its potential energy. The
latter comes about from its position in the electric field. Only when in the
sheath will the electron have any potential energy.

The kinetic energy is given by

(4- 6)

The potential energy can be found
by integrating the force on the electron as it moves from its current position
to the sheath boundary, z = s.

(4- 7)

We can have Mathematica do this
integration, using the Integrate function:

The results can be simplified with
the Collect function:

The expression can be made simpler
still by writing it in terms of E instead of V. This can be done by
taking our expression for Ez, solving it for Vd, and substituting that
expression into the potential energy expression above. An additional
simplification using the Apart and Simplify functions gets us to the final form:

Thus the potential energy can be
simply written as

(4- 8)

This, combined with the kinetic
energy, can be used to plot the total energy, as shown in Code 4- 7.

In Figure 4- 6, we can see that the
total energy is conserved. This confirms that NDSolve is giving a
reasonable result. We can also plot the potential and kinetic energy individually
and see how the energy cycles back and forth between the two (Figure 4- 7
and Figure 4- 8). Each time the electron returns to the
target surface, all of its energy is converted into potential energy, and its
speed drops to zero.

After an electron is emitted from the target
surface, it quickly accelerates to high speeds. This can be seen in Figure
4- 9. With -300 V applied to the target, the electron
reaches speeds of over 8 x 106 m/s in less than one
nanosecond. While this is quite fast, it is still just a few percent of
the speed of light. That is what allows us to ignore relativistic effects
and use classical physics in solving for the electron trajectory.

Code 4- 8. A ParametricPlot is used to plot
electron position vs velocity.

One can make some very interesting figures by
plotting the phase space coordinates. For instance in Figure 4- 10,
the y component of velocity is plotted against the y
position. In Figure 4- 11, the same is done for the x
and z components.

A key aspect of magnetron
sputtering is the fact that electrons follow a closed loop as they move over
the target surface. This is due to their drift velocity. The first
thing to point out about the drift velocity is that different electrons have different
drift rates. Those in the center of the racetrack go much faster.
Those further away from the center line of the racetrack drift more slowly.
We can see that by launching some additional electrons.

In Figure 4- 12, we launch from three
different y positions, -0.007 m, -0.015 m, and -0.020 m, rerunning NDSolve for
0.1 ms for each case.

The inner trajectory is our original launch
point. By moving the launch point out to 0.015 m, the trajectory is
wider, and the drift velocity is much lower. Going further out to 0.020 m,
the electron is only weakly bound by the magnetic field and doesnÕt seem to
orbit the B field lines. This can be seen more clearly in a side view (Figure
4- 13).

This electron drift down the racetrack is generally
referred to as ExB drift. However, other mechanisms also contribute to the
drift. In addition to ExB drift, there is gradient drift and curvature
drift (Chen 1984). All three cause the electron to
move in the x direction. We can estimate the magnitude of these mechanisms
to get a feel for which dominates. The average drift velocity of the
electron launched from y = -0.07 m is

On average, the electron is moving at 1.4 x 106
m/s in the x direction. The ExB drift is given by

We can plot this for our electron
and see how it varies with position.

Code 4- 10. Position versus ExB drift
velocity.

Figure 4- 14. ExB drift
velocity.

As expected, the ExB drift velocity
can be quite high when the electron is in the sheath, reaching more than 6 x 106
m/s. But outside of the sheath it goes to zero. The average drift
velocity is calculated as follows:

Code 4- 11. Calculating average ExB
drift.

This is quite similar to the
average drift velocity seen by our electron, which suggests ExB drift is the
dominant mechanism. This is consistent with experimental measurements of
Bradley, et al. (Bradley, Thompson and Gonzalvo 2001)

Even though the ExB drift drops to zero outside
the sheath, our plot of trajectory (Figure 4- 11) shows an
apparent drift in the x direction all of the time. This suggests that the
other drift mechanisms do play some role as well.

The other two sources of electron drift in a B
field are gradient drift and curvature drift. The grad B drift is given
by Chen (Chen 1984):

where
is the electron velocity perpendicular to the field and
is the Larmor radius given by

The curvature drift is given by

where
is the electron velocity parallel to the the B field and rc
is the radius of curvature of the B field.

The Grad B drift can be calculated with this
function:

Code 4- 12. A function for the Grad B drift
velocity.

Plotting this drift (Figure 4- 15)
we see that is it quite small compared to ExB drift. On average the
curvature drift is only

For the curvature drift, we need to estimate the
curvature of the B field. From Figure 4- 2 we can see that
for an electron starting at y = 0.007 m, the curvature is about 1.5 cm. A
more precise calculation shows the average curvature of the B field along this
trajectory to be 1.38 cm. The radius vector can then be written as

The peak curvature drift
is about half the size of the ExB drift. More importantly, this peak drift
occurs exactly when the ExB drift is zero. This can be seen by overlaying
the two plots:

Figure 4- 17. Curvature drift
overlaid on ExB drift.

The average curvature drift is

From the average drift values we can say that for
this case, the electron drift is due primarily to ExB, with small contributions
from Grad B drift and curvature drift. We made a few simplifying assumptions
and as a result the three mechanisms donÕt add up to the average drift velocity
calculated at the start of this section. Still, these calculations help
us understand the nature of the drift of electrons around the racetrack.

In practice it is difficult to
maintain a constant magnetic field strength around the entire racetrack.
In particular, the turnaround region typically has a weaker field. What
is the effect of magnetic field gradients on the electron trajectory?
Buyle et al. found that the height, width and velocity of the electron
trajectory all change as the electron transitioned from a region of weak field
to high field (Buyle, et al. 2004). We can do a
similar calculation by modifying our magpack list slightly:

We have taken each long magnet and
divided it in two, with a weak end and a strong end. Figure 4- 18
shows the B field in the target plane half way between the magnets.

In order to find the electron trajectory, we run
the NDSolve function like before. It takes a little longer to run because
our magpack has twice the magnets in it.

Code 4- 13. Solving electron equations
of motion.

The trajectory is shown in Figure 4- 19.
In the weak field region, the electron has a higher drift velocity, so the
pathline is more spread out. As the electron moves into the stronger
field, its drift velocity slows and the pattern becomes tighter. This
means that the electron spends less time in the weak field region. This
leads to less time ionizing argon atoms and thus a lower sputter rate in the
weak region.

A side view of the trajectories can be seen in Figure
4- 20. The particle initially follows an arc close to the
surface. Much of the time the electron is in the 1 mm sheath. Once
it transitions into the stronger field, the electron moves up to a higher B
field line. This both broadens its arc and gets it out of the sheath for
much of the time. Both of these have implications for sputtering.
The broader arc should result in a wider erosion groove in the target, boosting
target utilization. The higher trajectory means that more ions are formed
above the sheath. As they are attracted to the target, the ions fall
through the full sheath potential (300 V in this case), transmitting maximum
energy to the target. When ionization occurs inside the sheath, the ions
fall through only a portion of the sheath region and thus donÕt pick up the
full 300 eV of energy.

Figure 4- 20.
The electron move up to a higher trajectory as B field
strengthens.

We can easily try the reverse case,
where the electron starts in a strong field and transitions to a weak field.
The magpack then becomes

The trajectory plots are shown in Figure 4- 21.
In this case, the electron starts in a trajectory with a slow drift velocity
and then transitions into a faster one. The arc also goes in the opposite
direction. It starts high, and as the field weakens, the electron moves
down to a lower B field line.

Figure 4- 22. The weakening B
field forces the electrons closer to the target
surface.

Buyle (Buyle, et al. 2004)
used these results to explain the cross corner effect—that is the
observation of higher erosion rates just after the turnaround, at both ends of
the target. They noted that as the electron comes out of the turnaround
into a stronger field, it both slows down and drops to a lower orbit.
Both effects increase the electron density there, leading to more ionization
and higher erosion rates.

We can launch a particle from anywhere along the racetrack
and then solve for its position in the usual way. LetÕs see how the
motion evolves as the electron moves through the turnaround region.

Code 4- 15. NDSolve is used to find the
electron trajectory at the end of the magnet array.

As shown in Figure 4- 25, the
electron trajectory has three distinct motions. On a fine scale the
electron is circling around the local B field line. On a bigger scale, it
is following an arc-shaped path from the target surface upward and then back
down to the target. Lastly, it is following the ExB drift direction,
which takes it around the racetrack.

This last motion can be seen in Figure 4- 26
where the trajectory is superimposed on both the racetrack (dashed line) and
the contours of the parallel B field. There are several interesting
things to note in this figure. First, the electron position in the B
field shifts dramatically as it reaches the end of the magnet array. At the
start of its trajectory, the electron is primarily in the strongest part of the
B field (indicated by lighter shading). In the turnaround region, not
only is the field weaker overall, the electron drifts into an even weaker
portion of the field. As we saw earlier, this weaker B field results in a
faster ExB drift, leading to less time spent in the turnaround region and less
ionization and sputtering. So just from this plot we would expect the
target erosion to be less at the turnaround than in the straight section of the
magnet array.

Another interesting aspect of the trajectory is
the relative positions of the electron trajectory and the dashed Bz = 0
line. The general rule of thumb is that the racetrack will be centered
around this line. This is true when the field lines follow nice,
symmetric arcs, as in Figure 4- 2. However, the field lines
for this magnet array are not symmetric, as shown in Figure 4- 24.
At the target surface, the Bz = 0 point is closer to the outer magnets,
rather than centered between the inner and outer magnets. A slice through
the turnaround region (not plotted) would show the opposite trend. From this
we can conclude that the racetrack is only approximately located by the Bz = 0
line.

The electron motion around the field lines also
changes as the electron moves through the turnaround. The spirals become
larger and more closely spaced. This indicates that some of the electronÕs
kinetic energy has been diverted from ExB drift into spiral motion. We
can check this by finding the average velocity in the x direction at the start
and the end of the trajectory:

The electron has slowed by nearly 40 percent in
the x direction. This would suggest that the erosion rate could be higher
on the outboard side of the turnaround compared to the inboard side. This
cross corner effect, which is experimentally observed (Fan, Zhou
and Gracio 2003), is generally what limits target lifetimes.

Figure 4- 26.
Electron trajectory superimposed on racetrack (dashed line) and
contours of the B field parallel to the target.

We can look at a side view of the electron
trajectory to get further insights into the effect of the turnaround. Figure
4- 27 shows three planes along the electron path where we examine
the trajectory. These are the initial profile, the profile halfway
through the turnaround, and the final profile. The initial and final
profiles are compared in Figure 4- 28.

Figure 4- 27.
The rectangles indicate where we examine the trajectory in Figure 4- 28
and Figure 4- 29.

The initial profile shows a small amount of motion
around the B field lines as noted above. The dashed line shows the
profile at the end of the simulation. The orbits are much larger and the
width has increased slightly. This would suggest that the erosion groove will
be somewhat wider on the outbound side of the turnaround.

A comparison between the initial profile and the
midpoint profile is shown in Figure 4- 29. Here the width of
the arc has shrunk compared to the incoming profile, suggesting a narrower
erosion groove in the turnaround.

Figure 4- 29.
Electron path at the start and midpoint in its trajectory.

From the above results, we can see that even these
simple models of electron motion can provide useful insights into the behavior
of the magnetron. They allow us to better understand the nature of
electron drift as well as the effect of changing field strength on electron
motion. However, to go further, we need to include the effect of
electron-argon collisions. That is the subject to which we now turn.